Find three rational numbers between `(3)/(5) "and" (4)/(7)`.

Answer

To find three rational numbers between \( \frac{3}{5} \) and \( \frac{4}{7} \), we can follow these steps:

### Step 1: Find a common denominator
The denominators of the two fractions are 5 and 7. The least common multiple (LCM) of 5 and 7 is 35. We will convert both fractions to have this common denominator.

### Step 2: Convert \( \frac{3}{5} \) to the common denominator
To convert \( \frac{3}{5} \) to a denominator of 35, we multiply both the numerator and the denominator by 7:
\[ \frac{3}{5} = \frac{3 \times 7}{5 \times 7} = \frac{21}{35} \]

### Step 3: Convert \( \frac{4}{7} \) to the common denominator
Next, we convert \( \frac{4}{7} \) to a denominator of 35 by multiplying both the numerator and the denominator by 5:
\[ \frac{4}{7} = \frac{4 \times 5}{7 \times 5} = \frac{20}{35} \]

### Step 4: Identify the range
Now we have:
\[ \frac{21}{35} \quad \text{and} \quad \frac{20}{35} \]
However, we notice that \( \frac{21}{35} \) is greater than \( \frac{20}{35} \). We need to correct this. The correct order is:
\[ \frac{20}{35} < \frac{21}{35} \]

### Step 5: Find three rational numbers between \( \frac{20}{35} \) and \( \frac{21}{35} \)
To find three rational numbers between \( \frac{20}{35} \) and \( \frac{21}{35} \), we can take fractions with the same denominator, which is 35. We can choose numbers between 20 and 21.

The three rational numbers can be:
- \( \frac{201}{350} \)
- \( \frac{202}{350} \)
- \( \frac{203}{350} \)

### Final Answer
Thus, three rational numbers between \( \frac{3}{5} \) and \( \frac{4}{7} \) are:
1. \( \frac{201}{350} \)
2. \( \frac{202}{350} \)
3. \( \frac{203}{350} \)

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